Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-17T16:47:27.631Z Has data issue: false hasContentIssue false
  • English
  • Français

The inadequacy of Kripke's semantical analysis of D2 and D3

Published online by Cambridge University Press:  12 March 2014

R. Routley  and
H. Montgomery
R. Routley
Affiliation:
Monash University and University of Canterbury
H. Montgomery
Affiliation:
Monash University and University of Canterbury
Get access Rights & Permissions [Opens in a new window]

Extract

Kripke's model structure for D2 is a quadruple (G, K, R, N) where N is the set of normal elements, i.e., the subset of K such that for every H in N, HRH. (See [1, p. 220, p. 211].) In fact, however, this structure provides a model for E2, as the work of Lemmon shows ([2, pp. 58–62]).

To show the inadequacy of Kripke's model structure, we show that the E2 thesis □B ⊃ B, not a thesis of D2, is valid under Kripke's modelling for D2. Suppose, for a reductio argument, that □ B ⊃ B is false for some valuation ϕ in some D2 model structure. Thenyϕ(□B ⊃ B, G) = F; so by truth-functional assignments

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 33 , Issue 4 , January 1969 , pp. 568
Copyright
Copyright © Association for Symbolic Logic 1969

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Saul Kripke, A., Semantical analysis of modal logic. II: non-normal modal propositional calculi, Symposium on the theory of models, North-Holland, Amsterdam, 1965, pp. 206–220.Google Scholar
[2]Lemmon, E. J., Algebraic semantics for modal logics. I, this Journal, vol. 31 (1966), pp. 46–65.Google Scholar
[3]Lemmon, E. J., Algebraic semantics for modal logics. II, this Journal, vol. 31 (1966), pp. 191–218.Google Scholar